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Theorem lt6abl 18119

Description: A group with fewer than 6 elements is abelian. (Contributed by Mario Carneiro, 27-Apr-2016.)

**Hypothesis**
Ref	Expression
cygctb.1	⊢ 𝐵 = (Base‘𝐺)

**Assertion**
Ref	Expression
lt6abl	⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) < 6) → 𝐺 ∈ Abel)

Proof of Theorem lt6abl Dummy variables 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cygctb.1	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
2	1	grpbn0 17274	. . . . . 6 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
3	2	adantr 480	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) < 6) → 𝐵 ≠ ∅)
4		6re 10978	. . . . . . . 8 ⊢ 6 ∈ ℝ
5		rexr 9964	. . . . . . . 8 ⊢ (6 ∈ ℝ → 6 ∈ ℝ^*)
6		pnfnlt 11838	. . . . . . . 8 ⊢ (6 ∈ ℝ^* → ¬ +∞ < 6)
7	4, 5, 6	mp2b 10	. . . . . . 7 ⊢ ¬ +∞ < 6
8		fvex 6113	. . . . . . . . . . . . 13 ⊢ (Base‘𝐺) ∈ V
9	1, 8	eqeltri 2684	. . . . . . . . . . . 12 ⊢ 𝐵 ∈ V
10	9	a1i 11	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → 𝐵 ∈ V)
11		hashinf 12984	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ V ∧ ¬ 𝐵 ∈ Fin) → (#‘𝐵) = +∞)
12	10, 11	sylan 487	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ ¬ 𝐵 ∈ Fin) → (#‘𝐵) = +∞)
13	12	breq1d 4593	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ ¬ 𝐵 ∈ Fin) → ((#‘𝐵) < 6 ↔ +∞ < 6))
14	13	biimpd 218	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ ¬ 𝐵 ∈ Fin) → ((#‘𝐵) < 6 → +∞ < 6))
15	14	impancom 455	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) < 6) → (¬ 𝐵 ∈ Fin → +∞ < 6))
16	7, 15	mt3i 140	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) < 6) → 𝐵 ∈ Fin)
17		hashnncl 13018	. . . . . 6 ⊢ (𝐵 ∈ Fin → ((#‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
18	16, 17	syl 17	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) < 6) → ((#‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
19	3, 18	mpbird 246	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) < 6) → (#‘𝐵) ∈ ℕ)
20		nnuz 11599	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
21	19, 20	syl6eleq 2698	. . 3 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) < 6) → (#‘𝐵) ∈ (ℤ_≥‘1))
22		6nn 11066	. . . . 5 ⊢ 6 ∈ ℕ
23	22	nnzi 11278	. . . 4 ⊢ 6 ∈ ℤ
24	23	a1i 11	. . 3 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) < 6) → 6 ∈ ℤ)
25		simpr 476	. . 3 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) < 6) → (#‘𝐵) < 6)
26		elfzo2 12342	. . 3 ⊢ ((#‘𝐵) ∈ (1..^6) ↔ ((#‘𝐵) ∈ (ℤ_≥‘1) ∧ 6 ∈ ℤ ∧ (#‘𝐵) < 6))
27	21, 24, 25, 26	syl3anbrc 1239	. 2 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) < 6) → (#‘𝐵) ∈ (1..^6))
28		df-6 10960	. . . . . . 7 ⊢ 6 = (5 + 1)
29	28	oveq2i 6560	. . . . . 6 ⊢ (1..^6) = (1..^(5 + 1))
30	29	eleq2i 2680	. . . . 5 ⊢ ((#‘𝐵) ∈ (1..^6) ↔ (#‘𝐵) ∈ (1..^(5 + 1)))
31		5nn 11065	. . . . . . 7 ⊢ 5 ∈ ℕ
32	31, 20	eleqtri 2686	. . . . . 6 ⊢ 5 ∈ (ℤ_≥‘1)
33		fzosplitsni 12444	. . . . . 6 ⊢ (5 ∈ (ℤ_≥‘1) → ((#‘𝐵) ∈ (1..^(5 + 1)) ↔ ((#‘𝐵) ∈ (1..^5) ∨ (#‘𝐵) = 5)))
34	32, 33	ax-mp 5	. . . . 5 ⊢ ((#‘𝐵) ∈ (1..^(5 + 1)) ↔ ((#‘𝐵) ∈ (1..^5) ∨ (#‘𝐵) = 5))
35	30, 34	bitri 263	. . . 4 ⊢ ((#‘𝐵) ∈ (1..^6) ↔ ((#‘𝐵) ∈ (1..^5) ∨ (#‘𝐵) = 5))
36		df-5 10959	. . . . . . . . 9 ⊢ 5 = (4 + 1)
37	36	oveq2i 6560	. . . . . . . 8 ⊢ (1..^5) = (1..^(4 + 1))
38	37	eleq2i 2680	. . . . . . 7 ⊢ ((#‘𝐵) ∈ (1..^5) ↔ (#‘𝐵) ∈ (1..^(4 + 1)))
39		4nn 11064	. . . . . . . . 9 ⊢ 4 ∈ ℕ
40	39, 20	eleqtri 2686	. . . . . . . 8 ⊢ 4 ∈ (ℤ_≥‘1)
41		fzosplitsni 12444	. . . . . . . 8 ⊢ (4 ∈ (ℤ_≥‘1) → ((#‘𝐵) ∈ (1..^(4 + 1)) ↔ ((#‘𝐵) ∈ (1..^4) ∨ (#‘𝐵) = 4)))
42	40, 41	ax-mp 5	. . . . . . 7 ⊢ ((#‘𝐵) ∈ (1..^(4 + 1)) ↔ ((#‘𝐵) ∈ (1..^4) ∨ (#‘𝐵) = 4))
43	38, 42	bitri 263	. . . . . 6 ⊢ ((#‘𝐵) ∈ (1..^5) ↔ ((#‘𝐵) ∈ (1..^4) ∨ (#‘𝐵) = 4))
44		df-4 10958	. . . . . . . . . . 11 ⊢ 4 = (3 + 1)
45	44	oveq2i 6560	. . . . . . . . . 10 ⊢ (1..^4) = (1..^(3 + 1))
46	45	eleq2i 2680	. . . . . . . . 9 ⊢ ((#‘𝐵) ∈ (1..^4) ↔ (#‘𝐵) ∈ (1..^(3 + 1)))
47		3nn 11063	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ
48	47, 20	eleqtri 2686	. . . . . . . . . 10 ⊢ 3 ∈ (ℤ_≥‘1)
49		fzosplitsni 12444	. . . . . . . . . 10 ⊢ (3 ∈ (ℤ_≥‘1) → ((#‘𝐵) ∈ (1..^(3 + 1)) ↔ ((#‘𝐵) ∈ (1..^3) ∨ (#‘𝐵) = 3)))
50	48, 49	ax-mp 5	. . . . . . . . 9 ⊢ ((#‘𝐵) ∈ (1..^(3 + 1)) ↔ ((#‘𝐵) ∈ (1..^3) ∨ (#‘𝐵) = 3))
51	46, 50	bitri 263	. . . . . . . 8 ⊢ ((#‘𝐵) ∈ (1..^4) ↔ ((#‘𝐵) ∈ (1..^3) ∨ (#‘𝐵) = 3))
52		df-3 10957	. . . . . . . . . . . . 13 ⊢ 3 = (2 + 1)
53	52	oveq2i 6560	. . . . . . . . . . . 12 ⊢ (1..^3) = (1..^(2 + 1))
54	53	eleq2i 2680	. . . . . . . . . . 11 ⊢ ((#‘𝐵) ∈ (1..^3) ↔ (#‘𝐵) ∈ (1..^(2 + 1)))
55		2eluzge1 11610	. . . . . . . . . . . 12 ⊢ 2 ∈ (ℤ_≥‘1)
56		fzosplitsni 12444	. . . . . . . . . . . 12 ⊢ (2 ∈ (ℤ_≥‘1) → ((#‘𝐵) ∈ (1..^(2 + 1)) ↔ ((#‘𝐵) ∈ (1..^2) ∨ (#‘𝐵) = 2)))
57	55, 56	ax-mp 5	. . . . . . . . . . 11 ⊢ ((#‘𝐵) ∈ (1..^(2 + 1)) ↔ ((#‘𝐵) ∈ (1..^2) ∨ (#‘𝐵) = 2))
58	54, 57	bitri 263	. . . . . . . . . 10 ⊢ ((#‘𝐵) ∈ (1..^3) ↔ ((#‘𝐵) ∈ (1..^2) ∨ (#‘𝐵) = 2))
59		elsni 4142	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝐵) ∈ {1} → (#‘𝐵) = 1)
60		fzo12sn 12418	. . . . . . . . . . . . . . . . 17 ⊢ (1..^2) = {1}
61	59, 60	eleq2s 2706	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝐵) ∈ (1..^2) → (#‘𝐵) = 1)
62	61	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) ∈ (1..^2)) → (#‘𝐵) = 1)
63		hash1 13053	. . . . . . . . . . . . . . 15 ⊢ (#‘1_𝑜) = 1
64	62, 63	syl6eqr 2662	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) ∈ (1..^2)) → (#‘𝐵) = (#‘1_𝑜))
65		1nn0 11185	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℕ₀
66	62, 65	syl6eqel 2696	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) ∈ (1..^2)) → (#‘𝐵) ∈ ℕ₀)
67		hashclb 13011	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ V → (𝐵 ∈ Fin ↔ (#‘𝐵) ∈ ℕ₀))
68	9, 67	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ Fin ↔ (#‘𝐵) ∈ ℕ₀)
69	66, 68	sylibr 223	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) ∈ (1..^2)) → 𝐵 ∈ Fin)
70		1onn 7606	. . . . . . . . . . . . . . . 16 ⊢ 1_𝑜 ∈ ω
71		nnfi 8038	. . . . . . . . . . . . . . . 16 ⊢ (1_𝑜 ∈ ω → 1_𝑜 ∈ Fin)
72	70, 71	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 1_𝑜 ∈ Fin
73		hashen 12997	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ Fin ∧ 1_𝑜 ∈ Fin) → ((#‘𝐵) = (#‘1_𝑜) ↔ 𝐵 ≈ 1_𝑜))
74	69, 72, 73	sylancl 693	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) ∈ (1..^2)) → ((#‘𝐵) = (#‘1_𝑜) ↔ 𝐵 ≈ 1_𝑜))
75	64, 74	mpbid 221	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) ∈ (1..^2)) → 𝐵 ≈ 1_𝑜)
76	1	0cyg 18117	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ≈ 1_𝑜) → 𝐺 ∈ CycGrp)
77		cygabl 18115	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Abel)
78	76, 77	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ≈ 1_𝑜) → 𝐺 ∈ Abel)
79	75, 78	syldan 486	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) ∈ (1..^2)) → 𝐺 ∈ Abel)
80	79	ex 449	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → ((#‘𝐵) ∈ (1..^2) → 𝐺 ∈ Abel))
81		id 22	. . . . . . . . . . . . 13 ⊢ ((#‘𝐵) = 2 → (#‘𝐵) = 2)
82		2prm 15243	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℙ
83	81, 82	syl6eqel 2696	. . . . . . . . . . . 12 ⊢ ((#‘𝐵) = 2 → (#‘𝐵) ∈ ℙ)
84	1	prmcyg 18118	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) ∈ ℙ) → 𝐺 ∈ CycGrp)
85	84, 77	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) ∈ ℙ) → 𝐺 ∈ Abel)
86	85	ex 449	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Grp → ((#‘𝐵) ∈ ℙ → 𝐺 ∈ Abel))
87	83, 86	syl5 33	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → ((#‘𝐵) = 2 → 𝐺 ∈ Abel))
88	80, 87	jaod 394	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → (((#‘𝐵) ∈ (1..^2) ∨ (#‘𝐵) = 2) → 𝐺 ∈ Abel))
89	58, 88	syl5bi 231	. . . . . . . . 9 ⊢ (𝐺 ∈ Grp → ((#‘𝐵) ∈ (1..^3) → 𝐺 ∈ Abel))
90		id 22	. . . . . . . . . . 11 ⊢ ((#‘𝐵) = 3 → (#‘𝐵) = 3)
91		3prm 15244	. . . . . . . . . . 11 ⊢ 3 ∈ ℙ
92	90, 91	syl6eqel 2696	. . . . . . . . . 10 ⊢ ((#‘𝐵) = 3 → (#‘𝐵) ∈ ℙ)
93	92, 86	syl5 33	. . . . . . . . 9 ⊢ (𝐺 ∈ Grp → ((#‘𝐵) = 3 → 𝐺 ∈ Abel))
94	89, 93	jaod 394	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → (((#‘𝐵) ∈ (1..^3) ∨ (#‘𝐵) = 3) → 𝐺 ∈ Abel))
95	51, 94	syl5bi 231	. . . . . . 7 ⊢ (𝐺 ∈ Grp → ((#‘𝐵) ∈ (1..^4) → 𝐺 ∈ Abel))
96		simpl 472	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) → 𝐺 ∈ Grp)
97		2z 11286	. . . . . . . . . . 11 ⊢ 2 ∈ ℤ
98		eqid 2610	. . . . . . . . . . . 12 ⊢ (gEx‘𝐺) = (gEx‘𝐺)
99		eqid 2610	. . . . . . . . . . . 12 ⊢ (od‘𝐺) = (od‘𝐺)
100	1, 98, 99	gexdvds2 17823	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 2 ∈ ℤ) → ((gEx‘𝐺) ∥ 2 ↔ ∀𝑥 ∈ 𝐵 ((od‘𝐺)‘𝑥) ∥ 2))
101	96, 97, 100	sylancl 693	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) → ((gEx‘𝐺) ∥ 2 ↔ ∀𝑥 ∈ 𝐵 ((od‘𝐺)‘𝑥) ∥ 2))
102	1, 98	gex2abl 18077	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ (gEx‘𝐺) ∥ 2) → 𝐺 ∈ Abel)
103	102	ex 449	. . . . . . . . . . 11 ⊢ (𝐺 ∈ Grp → ((gEx‘𝐺) ∥ 2 → 𝐺 ∈ Abel))
104	103	adantr 480	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) → ((gEx‘𝐺) ∥ 2 → 𝐺 ∈ Abel))
105	101, 104	sylbird 249	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) → (∀𝑥 ∈ 𝐵 ((od‘𝐺)‘𝑥) ∥ 2 → 𝐺 ∈ Abel))
106		rexnal 2978	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ 𝐵 ¬ ((od‘𝐺)‘𝑥) ∥ 2 ↔ ¬ ∀𝑥 ∈ 𝐵 ((od‘𝐺)‘𝑥) ∥ 2)
107	96	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → 𝐺 ∈ Grp)
108		simprl 790	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → 𝑥 ∈ 𝐵)
109	1, 99	odcl 17778	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐵 → ((od‘𝐺)‘𝑥) ∈ ℕ₀)
110	109	ad2antrl 760	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → ((od‘𝐺)‘𝑥) ∈ ℕ₀)
111		4nn0 11188	. . . . . . . . . . . . . . . 16 ⊢ 4 ∈ ℕ₀
112	111	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → 4 ∈ ℕ₀)
113		simpr 476	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) → (#‘𝐵) = 4)
114	113, 111	syl6eqel 2696	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) → (#‘𝐵) ∈ ℕ₀)
115	114, 68	sylibr 223	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) → 𝐵 ∈ Fin)
116	115	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → 𝐵 ∈ Fin)
117	1, 99	oddvds2 17806	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin ∧ 𝑥 ∈ 𝐵) → ((od‘𝐺)‘𝑥) ∥ (#‘𝐵))
118	107, 116, 108, 117	syl3anc 1318	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → ((od‘𝐺)‘𝑥) ∥ (#‘𝐵))
119	113	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → (#‘𝐵) = 4)
120	118, 119	breqtrd 4609	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → ((od‘𝐺)‘𝑥) ∥ 4)
121		sq2 12822	. . . . . . . . . . . . . . . . 17 ⊢ (2↑2) = 4
122	97	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → 2 ∈ ℤ)
123		2nn0 11186	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℕ₀
124	123	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → 2 ∈ ℕ₀)
125	1, 99	odcl2 17805	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin ∧ 𝑥 ∈ 𝐵) → ((od‘𝐺)‘𝑥) ∈ ℕ)
126	107, 116, 108, 125	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → ((od‘𝐺)‘𝑥) ∈ ℕ)
127		pccl 15392	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2 ∈ ℙ ∧ ((od‘𝐺)‘𝑥) ∈ ℕ) → (2 pCnt ((od‘𝐺)‘𝑥)) ∈ ℕ₀)
128	82, 126, 127	sylancr 694	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → (2 pCnt ((od‘𝐺)‘𝑥)) ∈ ℕ₀)
129	128	nn0zd 11356	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → (2 pCnt ((od‘𝐺)‘𝑥)) ∈ ℤ)
130		df-2 10956	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 = (1 + 1)
131		simprr 792	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → ¬ ((od‘𝐺)‘𝑥) ∥ 2)
132		dvdsexp 14887	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((2 ∈ ℤ ∧ (2 pCnt ((od‘𝐺)‘𝑥)) ∈ ℕ₀ ∧ 1 ∈ (ℤ_≥‘(2 pCnt ((od‘𝐺)‘𝑥)))) → (2↑(2 pCnt ((od‘𝐺)‘𝑥))) ∥ (2↑1))
133	132	3expia 1259	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((2 ∈ ℤ ∧ (2 pCnt ((od‘𝐺)‘𝑥)) ∈ ℕ₀) → (1 ∈ (ℤ_≥‘(2 pCnt ((od‘𝐺)‘𝑥))) → (2↑(2 pCnt ((od‘𝐺)‘𝑥))) ∥ (2↑1)))
134	97, 128, 133	sylancr 694	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → (1 ∈ (ℤ_≥‘(2 pCnt ((od‘𝐺)‘𝑥))) → (2↑(2 pCnt ((od‘𝐺)‘𝑥))) ∥ (2↑1)))
135		1z 11284	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 1 ∈ ℤ
136		eluz 11577	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((2 pCnt ((od‘𝐺)‘𝑥)) ∈ ℤ ∧ 1 ∈ ℤ) → (1 ∈ (ℤ_≥‘(2 pCnt ((od‘𝐺)‘𝑥))) ↔ (2 pCnt ((od‘𝐺)‘𝑥)) ≤ 1))
137	129, 135, 136	sylancl 693	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → (1 ∈ (ℤ_≥‘(2 pCnt ((od‘𝐺)‘𝑥))) ↔ (2 pCnt ((od‘𝐺)‘𝑥)) ≤ 1))
138		oveq2 6557	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑛 = 2 → (2↑𝑛) = (2↑2))
139	138, 121	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑛 = 2 → (2↑𝑛) = 4)
140	139	breq2d 4595	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑛 = 2 → (((od‘𝐺)‘𝑥) ∥ (2↑𝑛) ↔ ((od‘𝐺)‘𝑥) ∥ 4))
141	140	rspcev 3282	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((2 ∈ ℕ₀ ∧ ((od‘𝐺)‘𝑥) ∥ 4) → ∃𝑛 ∈ ℕ₀ ((od‘𝐺)‘𝑥) ∥ (2↑𝑛))
142	123, 120, 141	sylancr 694	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → ∃𝑛 ∈ ℕ₀ ((od‘𝐺)‘𝑥) ∥ (2↑𝑛))
143		pcprmpw2 15424	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((2 ∈ ℙ ∧ ((od‘𝐺)‘𝑥) ∈ ℕ) → (∃𝑛 ∈ ℕ₀ ((od‘𝐺)‘𝑥) ∥ (2↑𝑛) ↔ ((od‘𝐺)‘𝑥) = (2↑(2 pCnt ((od‘𝐺)‘𝑥)))))
144	82, 126, 143	sylancr 694	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → (∃𝑛 ∈ ℕ₀ ((od‘𝐺)‘𝑥) ∥ (2↑𝑛) ↔ ((od‘𝐺)‘𝑥) = (2↑(2 pCnt ((od‘𝐺)‘𝑥)))))
145	142, 144	mpbid 221	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → ((od‘𝐺)‘𝑥) = (2↑(2 pCnt ((od‘𝐺)‘𝑥))))
146	145	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → (2↑(2 pCnt ((od‘𝐺)‘𝑥))) = ((od‘𝐺)‘𝑥))
147		2cn 10968	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 2 ∈ ℂ
148		exp1 12728	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (2 ∈ ℂ → (2↑1) = 2)
149	147, 148	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (2↑1) = 2
150	149	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → (2↑1) = 2)
151	146, 150	breq12d 4596	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → ((2↑(2 pCnt ((od‘𝐺)‘𝑥))) ∥ (2↑1) ↔ ((od‘𝐺)‘𝑥) ∥ 2))
152	134, 137, 151	3imtr3d 281	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → ((2 pCnt ((od‘𝐺)‘𝑥)) ≤ 1 → ((od‘𝐺)‘𝑥) ∥ 2))
153	131, 152	mtod 188	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → ¬ (2 pCnt ((od‘𝐺)‘𝑥)) ≤ 1)
154		1re 9918	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ∈ ℝ
155	128	nn0red 11229	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → (2 pCnt ((od‘𝐺)‘𝑥)) ∈ ℝ)
156		ltnle 9996	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 ∈ ℝ ∧ (2 pCnt ((od‘𝐺)‘𝑥)) ∈ ℝ) → (1 < (2 pCnt ((od‘𝐺)‘𝑥)) ↔ ¬ (2 pCnt ((od‘𝐺)‘𝑥)) ≤ 1))
157	154, 155, 156	sylancr 694	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → (1 < (2 pCnt ((od‘𝐺)‘𝑥)) ↔ ¬ (2 pCnt ((od‘𝐺)‘𝑥)) ≤ 1))
158	153, 157	mpbird 246	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → 1 < (2 pCnt ((od‘𝐺)‘𝑥)))
159		nn0ltp1le 11312	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1 ∈ ℕ₀ ∧ (2 pCnt ((od‘𝐺)‘𝑥)) ∈ ℕ₀) → (1 < (2 pCnt ((od‘𝐺)‘𝑥)) ↔ (1 + 1) ≤ (2 pCnt ((od‘𝐺)‘𝑥))))
160	65, 128, 159	sylancr 694	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → (1 < (2 pCnt ((od‘𝐺)‘𝑥)) ↔ (1 + 1) ≤ (2 pCnt ((od‘𝐺)‘𝑥))))
161	158, 160	mpbid 221	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → (1 + 1) ≤ (2 pCnt ((od‘𝐺)‘𝑥)))
162	130, 161	syl5eqbr 4618	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → 2 ≤ (2 pCnt ((od‘𝐺)‘𝑥)))
163		eluz2 11569	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2 pCnt ((od‘𝐺)‘𝑥)) ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ (2 pCnt ((od‘𝐺)‘𝑥)) ∈ ℤ ∧ 2 ≤ (2 pCnt ((od‘𝐺)‘𝑥))))
164	122, 129, 162, 163	syl3anbrc 1239	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → (2 pCnt ((od‘𝐺)‘𝑥)) ∈ (ℤ_≥‘2))
165		dvdsexp 14887	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 ∈ ℤ ∧ 2 ∈ ℕ₀ ∧ (2 pCnt ((od‘𝐺)‘𝑥)) ∈ (ℤ_≥‘2)) → (2↑2) ∥ (2↑(2 pCnt ((od‘𝐺)‘𝑥))))
166	122, 124, 164, 165	syl3anc 1318	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → (2↑2) ∥ (2↑(2 pCnt ((od‘𝐺)‘𝑥))))
167	121, 166	syl5eqbrr 4619	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → 4 ∥ (2↑(2 pCnt ((od‘𝐺)‘𝑥))))
168	167, 145	breqtrrd 4611	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → 4 ∥ ((od‘𝐺)‘𝑥))
169		dvdseq 14874	. . . . . . . . . . . . . . 15 ⊢ (((((od‘𝐺)‘𝑥) ∈ ℕ₀ ∧ 4 ∈ ℕ₀) ∧ (((od‘𝐺)‘𝑥) ∥ 4 ∧ 4 ∥ ((od‘𝐺)‘𝑥))) → ((od‘𝐺)‘𝑥) = 4)
170	110, 112, 120, 168, 169	syl22anc 1319	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → ((od‘𝐺)‘𝑥) = 4)
171	170, 119	eqtr4d 2647	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → ((od‘𝐺)‘𝑥) = (#‘𝐵))
172	1, 99, 107, 108, 171	iscygodd 18113	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → 𝐺 ∈ CycGrp)
173	172, 77	syl 17	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) ∧ (𝑥 ∈ 𝐵 ∧ ¬ ((od‘𝐺)‘𝑥) ∥ 2)) → 𝐺 ∈ Abel)
174	173	rexlimdvaa 3014	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) → (∃𝑥 ∈ 𝐵 ¬ ((od‘𝐺)‘𝑥) ∥ 2 → 𝐺 ∈ Abel))
175	106, 174	syl5bir 232	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) → (¬ ∀𝑥 ∈ 𝐵 ((od‘𝐺)‘𝑥) ∥ 2 → 𝐺 ∈ Abel))
176	105, 175	pm2.61d 169	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) = 4) → 𝐺 ∈ Abel)
177	176	ex 449	. . . . . . 7 ⊢ (𝐺 ∈ Grp → ((#‘𝐵) = 4 → 𝐺 ∈ Abel))
178	95, 177	jaod 394	. . . . . 6 ⊢ (𝐺 ∈ Grp → (((#‘𝐵) ∈ (1..^4) ∨ (#‘𝐵) = 4) → 𝐺 ∈ Abel))
179	43, 178	syl5bi 231	. . . . 5 ⊢ (𝐺 ∈ Grp → ((#‘𝐵) ∈ (1..^5) → 𝐺 ∈ Abel))
180		id 22	. . . . . . 7 ⊢ ((#‘𝐵) = 5 → (#‘𝐵) = 5)
181		5prm 15653	. . . . . . 7 ⊢ 5 ∈ ℙ
182	180, 181	syl6eqel 2696	. . . . . 6 ⊢ ((#‘𝐵) = 5 → (#‘𝐵) ∈ ℙ)
183	182, 86	syl5 33	. . . . 5 ⊢ (𝐺 ∈ Grp → ((#‘𝐵) = 5 → 𝐺 ∈ Abel))
184	179, 183	jaod 394	. . . 4 ⊢ (𝐺 ∈ Grp → (((#‘𝐵) ∈ (1..^5) ∨ (#‘𝐵) = 5) → 𝐺 ∈ Abel))
185	35, 184	syl5bi 231	. . 3 ⊢ (𝐺 ∈ Grp → ((#‘𝐵) ∈ (1..^6) → 𝐺 ∈ Abel))
186	185	imp 444	. 2 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) ∈ (1..^6)) → 𝐺 ∈ Abel)
187	27, 186	syldan 486	1 ⊢ ((𝐺 ∈ Grp ∧ (#‘𝐵) < 6) → 𝐺 ∈ Abel)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ∅c0 3874 {csn 4125 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ωcom 6957 1_𝑜c1o 7440 ≈ cen 7838 Fincfn 7841 ℂcc 9813 ℝcr 9814 1c1 9816 + caddc 9818 +∞cpnf 9950 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 ℕcn 10897 2c2 10947 3c3 10948 4c4 10949 5c5 10950 6c6 10951 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ..^cfzo 12334 ↑cexp 12722 #chash 12979 ∥ cdvds 14821 ℙcprime 15223 pCnt cpc 15379 Basecbs 15695 Grpcgrp 17245 odcod 17767 gExcgex 17768 Abelcabl 18017 CycGrpccyg 18102

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-er 7629 df-ec 7631 df-qs 7635 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-dvds 14822 df-gcd 15055 df-prm 15224 df-pc 15380 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-eqg 17416 df-od 17771 df-gex 17772 df-cmn 18018 df-abl 18019 df-cyg 18103

This theorem is referenced by: pgrple2abl 41940

W3C validator